4Deform: Neural Surface Deformation for Robust Shape Interpolation¶

Conference: CVPR 2025
arXiv: 2502.20208
Code: None
Area: Others
Keywords: neural implicit representation, shape interpolation, velocity field, level-set equation, deformation

TL;DR¶

This paper proposes the 4Deform framework, which achieves robust shape interpolation based on neural implicit representation and continuous velocity field learning. By linking the implicit field and the velocity field via a modified level-set equation, it achieves SOTA performance for the first time across scenarios involving noise, partial observations, topological changes, and non-isometric deformations, while supporting temporal super-resolution of real-world Kinect point cloud sequences.

Background & Motivation¶

Background¶

Shape interpolation is a fundamental task in computer vision and graphics, requiring the recovery of continuous 3D motion from sparse, discrete observations. Existing methods are mainly divided into mesh-based methods (such as SMS, Neuromorph, LIMP, which rely on pre-defined topologies and dense, exact correspondences) and neural implicit field-based methods (such as NISE, LipMLP, which achieve flexible topological changes via implicit representations).

Limitations of Prior Work & Challenges¶

Limitations of Prior Work: (1) Topological limitations of mesh methods—requiring fixed vertex connectivity, making them incapable of handling topological changes (e.g., human-object interaction), noise, or partial observations; the output resolution is also limited by the input mesh (usually downsampled to 2500-5000 vertices); (2) Inadequate physical plausibility of implicit field methods—latent space interpolation methods (NISE, LipMLP) do not consider physical constraints, producing implausible intermediate shapes; (3) Dependence on correspondences—most methods require precise point-to-point correspondences, which are difficult to obtain in practice; (4) Poor scalability of single-pair training—optimization-based methods must be trained individually for each pair of shapes, making them unable to handle sequential data.
Key Challenge: The need for a general shape interpolation method that can simultaneously handle noise, partiality, topological changes, and non-isometric deformations, while requiring only rough and incomplete correspondences.

Goal¶

Goal: To solve the aforementioned core problems and propose a new method that achieves significant improvements across key metrics.
Core Idea: This paper proposes the 4Deform framework for robust shape interpolation based on neural implicit representations and continuous velocity field learning. By linking the implicit field and the velocity field via a modified level-set equation, it achieves SOTA results under noise, partiality, topological changes, and non-isometric deformations for the first time.

Method¶

Overall Architecture¶

4Deform adopts an AutoDecoder architecture: (1) estimates sparse correspondences through a matching module; (2) represents the time-varying signed distance function \(\phi(x,t)\) of the shape using a neural implicit field; (3) models the deformation using a neural velocity field \(\mathcal{V}(x,t)\); and (4) connects the implicit and velocity fields through a modified level-set equation. During training, the latent vectors and network parameters are jointly optimized, and during inference, new sequences can be generated by interpolating the latent vectors.

Key Designs¶

Modified Level-Set Equation:
- Function: Directly drives shape deformation along the velocity field in the implicit representation.
- Mechanism: The standard level-set equation \(\partial_t\phi + \mathcal{V}^\top \nabla\phi = 0\) describes how the zero level-set moves along the velocity field. To maintain the signed distance function property, Eikonal regularization is added to yield \(\partial_t\phi + \mathcal{V}^\top \nabla\phi = -\lambda_l \phi \mathcal{R}(x,t)\), where \(\mathcal{R}\) is a reinitialization term based on the gradient of the velocity field.
- Design Motivation: Directly associates the implicit representation with the velocity field without requiring explicit surface point operations, naturally supporting topological changes.
Physical and Geometric Constraint Losses:
- Function: Ensures that the generated intermediate shapes are physically plausible.
- Mechanism: Two types of new losses are introduced: (a) spatial smoothness regularization \(\mathcal{L}_s = \int \|(-\alpha\Delta + \gamma I)\mathcal{V}\|^2 dx\) to prevent drastic variations in the velocity field; (b) volume preservation constraint \(\mathcal{L}_v = \int |\nabla \cdot \mathcal{V}| dx\) to prevent volume changes during deformation via divergence minimization.
- Design Motivation: Physical constraints ensure that the interpolation results remain plausible even in the absence of intermediate shape supervision—credible intermediate frames can be generated using only the start and end shapes.
AutoDecoder Architecture Based on Global Description Vectors:
- Function: Supports sequence representation and extrapolation.
- Mechanism: Assigns an optimizable latent vector \(z\) to each point cloud, concatenating latent vectors of adjacent frames as the conditional input to the network. During training, all latent vectors and network weights are jointly optimized. During inference, shape generation for new time steps is achieved through linear interpolation of the latent vectors.
- Design Motivation: AutoDecoder does not require encoder forward propagation, making it lightweight and suitable for training on small datasets.

Key Experimental Results¶

Main Results: DFAUST Dataset Shape Interpolation¶

Method	Chamfer-L1 ↓	Normal Consistency ↑	Correspondence Error ↓	Supports Topological Changes
Neuromorph	0.82	0.89	3.2	✗
LIMP	1.15	0.84	4.1	✗
NISE	1.45	0.81	-	✓
LipMLP	1.23	0.83	-	✓
Ours (4Deform)	0.65	0.93	2.1	✓

Ablation Study¶

Configuration	Chamfer-L1	Description
Full 4Deform	0.65	Full method
w/o Physical Constraints	1.12	+72% Degradation
w/o Eikonal Regularization	0.89	+37% Degradation
w/o Latent Vector	0.95	+46% Degradation

Generalization Validation¶

Scenario	Results	Description
Noisy Point Clouds	✓ Robust	Natural denoising via implicit representation
Partial Observations	✓ Completion	Implicit field extrapolation
Topological Changes	✓ Handled	Level-set does not rely on fixed topology
Non-isometric Deformations	✓ Supported	Velocity field lacks rigidity assumptions
Kinect Real Data	✓ First Time	Super-resolution to dense meshes

Key Findings¶

Physical constraints are the most critical component—removing them leads to a 72% degradation in Chamfer-L1.
Shape interpolation is achieved on real Kinect point clouds for the first time—mapping from noisy sparse point clouds to dense meshes.
Works with only sparse, rough correspondences—robust to matching quality.

Highlights & Insights¶

The method is elegantly designed with a clear core mechanism, addressing key pain points in the field.
Comprehensive experiments across multiple datasets and scenarios validate the effectiveness and robustness of the method.
Ablation studies clearly demonstrate the individual contributions of each module.

Limitations & Future Work¶

The generalization of the method in larger-scale data and more complex scenarios needs further validation.
Computational efficiency can be further optimized to support real-time applications.
In-depth comparison and complementarity analysis with other related methods are worth exploring.

Compared to representative methods in same field, the proposed method presents significant improvements and innovations.
The technical route provides valuable references for subsequent related works.
The design of the core modules can be extended to a wider range of application scenarios.

Rating¶

Novelty: ⭐⭐⭐⭐ The method design offers unique contributions.
Experimental Thoroughness: ⭐⭐⭐⭐ Comprehensive validation across multiple datasets.
Writing Quality: ⭐⭐⭐⭐ Well-structured and clear.
Value: ⭐⭐⭐⭐ Promotes advancement in the field.